Gibbs measures asymptotics

Abstract

Let (Ω, B, ν) be a measure space and H:Ω→ R⁺ be B measurable. Let ∫ _Ωe^-Hdν < ∞. For 0 < T < 1 let μ_H,T (·) be the probability measure defined by μ_H,T (A) = (∫_A e^-H/T dν)/(∫_Ω e^-H/T dν ), A ∈ B. In this paper, we study the behavior of μ_H,T (·) as T ↓ 0 and extend the results of Hwang (1980, 1981). When Ω is R and H achieves its minimum at a single value x0 (single well case) and H(·) is Holder continuous at x0 of order α, it is shown that if X_T is a random variable with probability distribution μ_H,T (·) then as T ↓ 0, i) X_T→ x0 in probability; ii) (X_t - x0)T^-1/α converges in distribution to an absolutely continuous symmetric distribution with density proportional to e^-c_α|x|α for some 0 < c_α < ∞. This is extended to the case when H achieves its minimum at a finite number of points (multiple well case). An extension of these results to the case H : Rⁿ→ R⁺ is also outlined.